In the 50s, when my mum would take my brother and myself for a day on the beach, her favourite billet was by the 'rails' which were roughly situated half way between what is now the Cat and Dog Home and the Marine Gardens.
Was this a leftover from a wartime launching facility? If so, what exactly?

You may be thinking about the place where they stored the Power Station's DKUW road/sea vehicle which was used to go out to the towers which stood about opposite the bottom of Kings Road. I thing the towers were outlets for cooling water

I remember the amphibian 'duck' well, it would drive out from the Power Station gate at the foot of Kings Road and continue down the ramp at the wind break structure on the prom, I haven't a clue what the proper architectural term is!My very earliest memories of the launching rails further along would be around 1954 ish and they were very derelict even then. I'll go with wangi's assertion that they were connected with landing craft launching.